Theorem carden2b 10007

Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 10006 are meant to replace carden 10591 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
carden2b	⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Proof of Theorem carden2b

Step	Hyp	Ref	Expression
1		cardne 10005	. . . . 5 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2		ennum 9987	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
3	2	biimpa 476	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4		cardid2 9993	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6		ensym 9043	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
7	6	adantr 480	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ≈ 𝐴)
8		entr 9046	. . . . . 6 ⊢ (((card‘𝐵) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (card‘𝐵) ≈ 𝐴)
9	5, 7, 8	syl2anc 584	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
10	1, 9	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11		cardon 9984	. . . . 5 ⊢ (card‘𝐴) ∈ On
12		cardon 9984	. . . . 5 ⊢ (card‘𝐵) ∈ On
13		ontri1 6418	. . . . 5 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
14	11, 12, 13	mp2an 692	. . . 4 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
15	10, 14	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16		cardne 10005	. . . . 5 ⊢ ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17		cardid2 9993	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18		id 22	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≈ 𝐵)
19		entr 9046	. . . . . 6 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (card‘𝐴) ≈ 𝐵)
20	17, 18, 19	syl2anr 597	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
21	16, 20	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22		ontri1 6418	. . . . 5 ⊢ (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
23	12, 11, 22	mp2an 692	. . . 4 ⊢ ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
24	21, 23	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
25	15, 24	eqssd 4001	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26		ndmfv 6941	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
27	26	adantl 481	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
28	2	notbid 318	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
29	28	biimpa 476	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30		ndmfv 6941	. . . 4 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
31	29, 30	syl 17	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
32	27, 31	eqtr4d 2780	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
33	25, 32	pm2.61dan 813	1 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143  dom cdm 5685  Oncon0 6384  ‘cfv 6561   ≈ cen 8982  cardccrd 9975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-er 8745  df-en 8986  df-card 9979

This theorem is referenced by:  card1  10008  carddom2  10017  cardennn  10023  cardsucinf  10024  pm54.43lem  10040  nnadju  10238  nnadjuALT  10239  ficardun  10241  ackbij1lem5  10263  ackbij1lem8  10266  ackbij1lem9  10267  ackbij2lem2  10279  carden  10591  r1tskina  10822  cardfz  14011